1Computer Graphics Implementation III Lecture 17 John Shearer Culture Lab – space 2

2Computer Graphics Objectives Survey Line Drawing Algorithms –DDA –Bresenham

3Computer Graphics Rasterization Rasterization (scan conversion) –Determine which pixels that are inside primitive specified by a set of vertices –Produces a set of fragments –Fragments have a location (pixel location) and other attributes such color and texture coordinates that are determined by interpolating values at vertices Pixel colors determined later using color, texture, and other vertex properties

4Computer Graphics Scan Conversion of Line Segments Start with line segment in window coordinates with integer values for endpoints Assume implementation has a write_pixel function y = mx + h

5Computer Graphics DDA Algorithm Digital Differential Analyzer –DDA was a mechanical device for numerical solution of differential equations –Line y=mx+ h satisfies differential equation dy/dx = m =  y/  x = y 2 -y 1 /x 2 -x 1 Along scan line  x = 1 for(x=x1; x<=x2,ix++) { y+=m;// Floating point addition write_pixel(x, round(y), line_color) }

6Computer Graphics Problem DDA = for each x plot pixel at closest y –Problems for steep lines

7Computer Graphics Using Symmetry Use for 1  m  0 For m > 1, swap role of x and y –For each y, plot closest x

8Computer Graphics Bresenham’s Algorithm DDA requires one floating point addition per step We can eliminate all fp through Bresenham’s algorithm Consider only 1  m  0 –Other cases by symmetry Assume pixel centers are at half integers If we start at a pixel that has been written, there are only two candidates for the next pixel to be written into the frame buffer

9Computer Graphics Candidate Pixels 1  m  0 last pixel candidates Note that line could have passed through any part of this pixel

10Computer Graphics Decision Variable d =  x(b-a) d is an integer d > 0 use upper pixel d < 0 use lower pixel

11Computer Graphics Incremental Form More efficient if we look at d k, the value of the decision variable at x = k dk+1= dk –2Dy, if dk <0 dk+1= dk –2(Dy- Dx), otherwise For each x, we need do only an integer addition and a test Single instruction on graphics chips

12Computer Graphics Polygon Scan Conversion Scan Conversion = Fill How to tell inside from outside –Convex easy –Nonsimple difficult –Odd even test Count edge crossings odd-even fill

13Computer Graphics Filling in the Frame Buffer Fill at end of pipeline –Convex Polygons only –Nonconvex polygons assumed to have been tessellated –Shades (colors) have been computed for vertices (Gouraud shading) –Combine with z-buffer algorithm March across scan lines interpolating shades Incremental work small

14Computer Graphics Using Interpolation C 1 C 2 C 3 specified by glColor or by vertex shading C 4 determined by interpolating between C 1 and C 2 C 5 determined by interpolating between C 2 and C 3 interpolate between C 4 and C 5 along span span C1C1 C3C3 C2C2 C5C5 C4C4 scan line

15Computer Graphics Aliasing Ideal rasterized line should be 1 pixel wide Choosing best y for each x (or visa versa) produces aliased raster lines

16Computer Graphics Antialiasing by Area Averaging Color multiple pixels for each x depending on coverage by ideal line original antialiased magnified

17Computer Graphics Polygon Aliasing Aliasing problems can be serious for polygons –Jaggedness of edges –Small polygons neglected –Need compositing so color of one polygon does not totally determine color of pixel All three polygons should contribute to color

18Computer Graphics

19Computer Graphics